Global finite dimensional flows (Q1906664)

This paper deals with the existence of global flows with singular vector fields on \(\mathbb{R}^n\) and gives conditions under which these flows avoid the set of singularities. The main concept is that of divergence with respect to a reference measure \(\sigma = e^w dL\), \(w \in C^1\), where \(dL\) denotes the Lebesgue measure in \(\mathbb{R}^n\). The divergence of a vector field \(B \in C^3 (\mathbb{R}^n, \mathbb{R}^n)\) with respect to the measure \(\sigma\) is defined as the element \(\delta_\sigma \in L^2_\sigma (\mathbb{R}^n)\) that satisfies the integral equality \[ \int \delta_\sigma Bu d \sigma = \int B \nabla u d \sigma, \quad u \in C^1_b (\mathbb{R}^n) \] where \(C^1_b (\mathbb{R}^n)\) denotes the class of \(C^1\) bounded real valued functions defined on \(\mathbb{R}^n\). It follows that \(\delta_\sigma B = \delta_L (B) - B \nabla w\). In the case where \(B\) is a smooth vector field, the author proves that if \(\delta_\sigma B \in L^\infty (\mathbb{R}^n)\) and \(B \in L^2_\sigma (\mathbb{R}^n)\), \(\int |x |^2d \sigma (x) < + \infty\), then for almost all initial conditions the flow associated with \(B\) is globally defined. In the case where \(B\) is singular on a closed set \(A \subset \mathbb{R}^n\), the author proves that if \(\delta_\sigma B \in L^\infty (\mathbb{R}^n)\), \(d_A \in W^{1,2}_\sigma (\mathbb{R}^n)\), \(B \in L^2_\sigma (\mathbb{R}^n)\), then for almost all initial conditions the solutions of the problem \(\dot x = B(x)\), \(x(0) = x_0\), \(x_0 \notin A\), avoid \(A\) (here \(d_A\) is the euclidean distance to \(A)\). As an example, the author considers in \(\mathbb{R}^{2n}\) a mechanical system characterized by a time independent function assumed to be smooth on \(\mathbb{R}^{2n}/A\), where \(A\) is any regular closed set in the phase space of the system.